Chapter 4 Lecture 3 Sections: 4.4 – 4.5

Multiplication Rule In our previous lecture, we focused on P(A or B), the probability that a trial has an outcome of A or B or both. We will now focus on finding the P(A and B), the probability that event A occurs in the first trial and the event that B occurs in the second trial and so on. P(A and B): P(event A occurs in the first trial and the event that B occurs in the second trial) Recall that we used addition for the P(A or B). We will now use multiplication for the P(A and B). Example: Say you flip a coin twice. What is the probability that you will get tails both times.

Illustration of the problem: T Thus we have T,T T,H H,T H,H 2 x 2 = 4 total outcomes. H T H H First toss, 2 possible outcomes Second toss, 2 possible outcomes in each

1. Say you take a quiz that consists of only two questions 1. Say you take a quiz that consists of only two questions. The first questions is True/False. The second questions is multiple choice with possible answers A, B, C, D. What is the probability that you get the two questions correct? 2. If you roll a die and flip a coin, what is the probability of getting Tails and rolling a 6. (T,6)?

3. A box contains 5 red markers and 7 black markers. We select 2 markers at random and find the probability of choosing a red marker first and a black marker second. P(red first)=5/12. What about P(black second)? This situation is just a little bit complicated. Remember that we are selecting markers from a box of 12, but we will need to take into consideration that we have already selected 1 marker and that marker happened to be red. In a situation like this, we will have to either a. Sample with replacement. b. Sample without replacement. a. Sample with replacement b. Sample without replacement

4. In a standard deck of cards, find the probability of selecting a heart first, a spade second and a club third. a. With replacement b. Without replacement One thing that you should notice in doing the problems with replacement is that they have one thing in common. The two events are independent of one another. The problems done without replacement are considered dependent. Independent Events: Two events A and B are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. If the events are not independent, they are said to be dependent.

Formulas for the Multiplication Rule: We will focus more on the formula for independent events. The formula for dependent events will be used when we discuss Conditional Probability.

6. A professor decides to give a 5 question multiple choice quiz consisting of A, B, C, D. Find the probability of getting the first 3 correct and the last two incorrect. 7. A space ship must be lifted into outer space with the assistance of four rocket boosters. If a rocket booster has a defect rate of 1.5%, what is the probability that the space ship will be lifted into outer space with the aid of four independent boosters?

8. A student in the class was born on October 31 8. A student in the class was born on October 31. What is the probability that two randomly selected students were born on the same day ignoring a leap year. 9. What is the probability of randomly selecting three people and they all have the same birth date?

Conditional Probability A Conditional Probability of an event is a probability obtained with the additional information that some other event has already occurred. P(B | A) denotes the Conditional Probability of event B occurring, given that event A has already occurred, and it can be found by the following formula. Our intuitive approach to conditional probability is a follows: The conditional probability of B given A can be found by assuming that event A has occurred and then working under that assumption to calculate the probability that event B will occur.

1. Back to our table Cancer Heart Disease Total Smoker 135 310 445 Nonsmoker 55 100 155 190 410 600 Find the probability that someone died of heart disease given that they were a smoker.

2. The table below summarizes results from a survey of 120 randomly selected college students. It states gender and political affiliation. Democrat Republican Total Male 13 27 40 Female 59 21 80 72 48 120 a. Find the probability that someone is a male given that they are a Democrat.

Complements & At Least One The Probability of “At Least One” is a procedure of calculating a probability that uses the multiplication rule and complements together. Key Vocabulary: At Least One = One or More The complement of At Least One = none Example: Recall the sample space of having 3 children BBB BBG BGB BGG GBB GBG GGB GGG At least one Boy The complement of At least one Boy=No Boys=All Girls

P(at least one of A) = 1–[P(Complement of A)]n = 1–P(none of A)]n To find the probability of At Least One of A, calculate the probability of the complement of A or none of A or raised to the number of trials, then subtract that result from 1. The formula is P(at least one of A) = 1–[P(Complement of A)]n = 1–P(none of A)]n where n is the number of trials Example: Let us find the probability of getting at least one boy among three births, but with out using the table. Which happens to be the same as using the table.

3. You take a 5 question True/False quiz 3. You take a 5 question True/False quiz. What is the probability of getting at least one correct on the quiz. 4. You take a 5 question multiple choice quiz (A,B,C,D). What is the probability of getting at least one correct on the quiz.

5. With one method of acceptance sampling, a sample of items is randomly selected and the entire batch is rejected if there is at least one defect. A company that manufactures satellite radios has just manufactured 5000 radios and the defect rate is 3%. If 10 of the radios are randomly selected and tested, what is the probability that the entire batch will be rejected?